Hello brilliant minds! Welcome to an exciting session where we're going to unravel one of the most fundamental yet intriguing concepts in Calculus: Continuity. Don't worry if it sounds intimidating; we'll start right from the very beginning, building our understanding brick by brick. By the end of this session, you'll not only understand what continuity means but also appreciate its significance in mathematics and the real world.

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### What Does "Continuity" Even Mean? - The Intuitive Idea

Imagine you're drawing a picture on a piece of paper. If you can draw an entire curve from one end to the other without ever lifting your pen, what would you call that curve? You'd probably say it's a "smooth" or "unbroken" curve, right? Well, in mathematics, we call such a curve continuous.

Think of it like a journey:
* If you're driving a car on a perfectly smooth road, without any bumps, sudden drops, or gaps, your journey is continuous.
* But what if you suddenly encounter a broken bridge (a gap), or a cliff (a sudden drop), or the road abruptly changes elevation (a jump)? Then your journey is interrupted, it's *discontinuous*.

In the world of functions, a function `f(x)` is said to be continuous at a point if its graph at that point doesn't have any breaks, jumps, or holes. It's like checking the road at a specific point – is it smooth there, or is there a problem?

Analogy Alert!
Think of a function's graph as a rollercoaster track.
* A continuous track means the ride is smooth; the car can go from one point to another without falling off or suddenly teleporting.
* A discontinuous track would mean there's a gap, a sudden drop, or a sharp, unbridgeable jump – not a fun ride!

So, at its heart, continuity is all about the "unbrokenness" of a function's graph.

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### Why Can't We Just "Look" at the Graph? - The Need for a Mathematical Definition

While the "no pen lift" analogy is great for intuition, it's not precise enough for mathematical analysis, especially when we can't always draw the graph or when we're dealing with complex functions. We need a rigorous, mathematical way to define continuity. This is where our good old friend, Limits, comes into play!

Remember limits? They tell us what value a function is *approaching* as `x` gets closer and closer to a certain point. Continuity combines this idea of 'approaching' with the actual 'value' at that point.

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### The Three Pillars of Continuity at a Point

For a function `f(x)` to be continuous at a point `x = a`, three crucial conditions must be met. Think of these as a checklist. If even one condition fails, the function is discontinuous at that point.

Here are the three conditions:

1. The function must be defined at `x = a`.
   
   This means that when you substitute `a` into `f(x)`, you should get a real, finite number. In other words, `f(a)` must exist. If `f(a)` is undefined (e.g., division by zero), then there's a hole or an asymptote at that point, making it discontinuous.
   
   
   
   *Visual:* Imagine a point `(a, f(a))` on the graph. If `f(a)` doesn't exist, there's literally no point on the graph at `x=a`.
   


2. The limit of the function must exist at `x = a`.
   
   This means that as `x` approaches `a` from the left side (LHL) and from the right side (RHL), the function `f(x)` must approach the *same* finite value. So, `lim (x→a⁻) f(x) = lim (x→a⁺) f(x) = L` (where `L` is a finite number). If the LHL and RHL are different, there's a "jump" in the graph, making it discontinuous.
   
   
   
   *Visual:* As you trace the graph towards `x=a` from both sides, your pen tips should be aiming for the same y-coordinate.
   


3. The limit value must be equal to the function value at `x = a`.
   
   This is the grand finale! It ties the previous two conditions together. It means that the value the function is *approaching* from both sides (`L`) must be exactly equal to the actual value of the function *at* that point (`f(a)`). So, `lim (x→a) f(x) = f(a)`.
   
   
   
   *Visual:* Not only are your pen tips aiming for the same y-coordinate from both sides, but there is also an actual point on the graph at `x=a` at exactly that y-coordinate. No holes, no jumps.
   

In short, for continuity at `x = a`:



`LHL = RHL = f(a)`



This single equation encapsulates all three conditions. If this equation holds true, the function is continuous at `x=a`.

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### Let's Put It Into Practice: Examples!

Understanding these conditions is one thing, but applying them is where the real learning happens. Let's work through some examples step-by-step.

#### Example 1: The "Always Continuous" Function (Polynomials)

Consider the function `f(x) = x^2`. Is it continuous at `x = 2`?

Step-by-step check:

1. Check if `f(2)` exists:
`f(2) = (2)^2 = 4`.
Yes, `f(2)` exists and is a finite number (4).

2. Check if `lim (x→2) f(x)` exists:
We need to check LHL and RHL.
* `LHL = lim (x→2⁻) x^2 = (2)^2 = 4`
* `RHL = lim (x→2⁺) x^2 = (2)^2 = 4`
Since LHL = RHL = 4, the limit exists and `lim (x→2) f(x) = 4`.

3. Check if `lim (x→2) f(x) = f(2)`:
From step 1, `f(2) = 4`.
From step 2, `lim (x→2) f(x) = 4`.
Since `4 = 4`, the condition is met!

Conclusion: Yes, `f(x) = x^2` is continuous at `x = 2`.
JEE/CBSE Focus: Polynomial functions like `x^2`, `3x+5`, `x^3-7x+1` are continuous everywhere, for all real numbers. This is a crucial property to remember!

#### Example 2: A Function with a Hole (Removable Discontinuity)

Consider the function `f(x) = (x^2 - 9) / (x - 3)`. Is it continuous at `x = 3`?

Step-by-step check:

1. Check if `f(3)` exists:
If we substitute `x = 3`, we get `f(3) = (3^2 - 9) / (3 - 3) = (9 - 9) / 0 = 0 / 0`.
This is an indeterminate form, meaning `f(3)` is undefined.

Since the first condition fails, we don't even need to check the others!

Conclusion: `f(x) = (x^2 - 9) / (x - 3)` is discontinuous at `x = 3`.
Important Note: This is a type of discontinuity called a removable discontinuity. If you simplify the function: `f(x) = (x-3)(x+3) / (x-3) = x+3` (for `x ≠ 3`), you'll see that the limit *would* exist, but the point itself is missing. This is like a tiny "hole" in the graph.

#### Example 3: A Piecewise Function (Jump Discontinuity)

Let's look at a piecewise function:
`f(x) = { x + 1, if x < 1 `
` { x - 1, if x ≥ 1 `

Is `f(x)` continuous at `x = 1`?

Step-by-step check:

1. Check if `f(1)` exists:
For `x = 1`, we use the second rule (`x - 1`).
`f(1) = 1 - 1 = 0`.
Yes, `f(1)` exists and is 0.

2. Check if `lim (x→1) f(x)` exists:
* LHL (as `x` approaches 1 from the left, `x < 1`): We use the first rule (`x + 1`).
`LHL = lim (x→1⁻) (x + 1) = 1 + 1 = 2`
* RHL (as `x` approaches 1 from the right, `x ≥ 1`): We use the second rule (`x - 1`).
`RHL = lim (x→1⁺) (x - 1) = 1 - 1 = 0`

Since `LHL ≠ RHL` (2 ≠ 0), the limit `lim (x→1) f(x)` does not exist.

Since the second condition fails, the function is discontinuous.

Conclusion: `f(x)` is discontinuous at `x = 1`.
Important Note: This is a jump discontinuity, because the function "jumps" from one value to another at `x = 1`.

#### Example 4: Making a Piecewise Function Continuous

Let's modify Example 3 slightly. Can we make this function continuous at `x = 1`?
`f(x) = { ax + 1, if x < 1 `
` { x - 1, if x ≥ 1 `

We want `f(x)` to be continuous at `x = 1`. This means we need `LHL = RHL = f(1)`.

1. Find `f(1)`:
Using the second rule (`x - 1` for `x ≥ 1`), `f(1) = 1 - 1 = 0`.

2. Find LHL:
Using the first rule (`ax + 1` for `x < 1`), `LHL = lim (x→1⁻) (ax + 1) = a(1) + 1 = a + 1`.

3. Find RHL:
Using the second rule (`x - 1` for `x ≥ 1`), `RHL = lim (x→1⁺) (x - 1) = 1 - 1 = 0`.

4. Equate them for continuity:
For continuity, `LHL = RHL = f(1)`.
So, `a + 1 = 0 = 0`.
From `a + 1 = 0`, we get `a = -1`.

Conclusion: If `a = -1`, the function `f(x)` will be continuous at `x = 1`.
JEE/CBSE Focus: This type of problem, where you need to find the value of a constant to make a function continuous, is very common in both Board exams and JEE Mains. It tests your understanding of the three conditions thoroughly.

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### Continuity on an Interval

So far, we've discussed continuity at a single point. But what if we want to talk about continuity over an entire range of `x` values? This leads us to continuity on an interval.

* A function `f(x)` is said to be continuous on an open interval (a, b) if it is continuous at every single point within that interval. (No pen lifting allowed anywhere between 'a' and 'b', excluding 'a' and 'b' themselves).

* A function `f(x)` is said to be continuous on a closed interval [a, b] if:
1. It is continuous on the open interval `(a, b)`.
2. It is right-continuous at `x = a`: This means `lim (x→a⁺) f(x) = f(a)`. (The graph starts exactly where it should, with no gap or jump immediately to the right of 'a').
3. It is left-continuous at `x = b`: This means `lim (x→b⁻) f(x) = f(b)`. (The graph ends exactly where it should, with no gap or jump immediately to the left of 'b').

Why separate conditions for endpoints?
At the endpoints `a` and `b`, you can only approach the point from one side while staying within the interval. For `a`, you can only approach from the right (`x→a⁺`), and for `b`, you can only approach from the left (`x→b⁻`). So, we adapt our limit condition accordingly.

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### Basic Properties of Continuous Functions

Continuous functions behave very nicely. Here are some fundamental properties:

Operation	Property
Sum/Difference	If `f(x)` and `g(x)` are continuous at `x=a`, then `(f ± g)(x)` is also continuous at `x=a`.
Product	If `f(x)` and `g(x)` are continuous at `x=a`, then `(f * g)(x)` is also continuous at `x=a`.
Quotient	If `f(x)` and `g(x)` are continuous at `x=a`, and `g(a) ≠ 0`, then `(f / g)(x)` is also continuous at `x=a`. (Watch out for division by zero!)
Scalar Multiple	If `f(x)` is continuous at `x=a` and `c` is a constant, then `(c * f)(x)` is also continuous at `x=a`.
Composition	If `f(x)` is continuous at `x=a` and `g(x)` is continuous at `f(a)`, then the composite function `(g o f)(x) = g(f(x))` is continuous at `x=a`.

These properties are incredibly useful because they allow us to deduce the continuity of complex functions from simpler ones without always going back to the three-condition checklist. For instance, since `x^2` and `sin(x)` are continuous everywhere, `x^2 + sin(x)` or `x^2 * sin(x)` will also be continuous everywhere.

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### Conclusion: Why is Continuity Important?

Understanding continuity is not just an academic exercise; it's a cornerstone for much of higher mathematics.

* In physics and engineering, many real-world phenomena (like temperature changes, motion, fluid flow) are modeled by continuous functions. If these functions were discontinuous, our models would break down, predicting impossible instantaneous changes.
* In calculus, concepts like differentiability (which we'll explore next!) and integrability heavily rely on the idea of continuity. You can't talk about the slope of a curve or the area under it if the curve itself has breaks or jumps.
* For JEE, a solid grasp of continuity is essential for solving problems involving limits, finding unknown constants in piecewise functions, and understanding the behavior of functions.

So, the next time you see a graph or a function, you'll have a keen eye to spot those crucial points where the "pen might have to be lifted"! Keep practicing these fundamental checks, and you'll master this concept in no time!

Welcome, future engineers, to a deep dive into one of the most fundamental concepts in calculus: Continuity. This isn't just a fancy word; it's a critical idea that underpins much of advanced mathematics, especially in topics like differentiability, integration, and even differential equations. For JEE aspirants, mastering continuity means not only understanding its definition but also being able to apply it robustly to various types of functions and complex scenarios.

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### 1. The Intuitive Idea of Continuity: No Breaks, No Jumps, No Holes!

Let's start with a very simple, visual understanding. Imagine you're drawing the graph of a function on a piece of paper. If you can draw the entire graph from left to right without ever lifting your pen from the paper, then that function is continuous over that path.

What prevents you from drawing without lifting your pen?
* A Jump: The graph suddenly leaps from one y-value to another at a certain x-value. Think of the cost of parking a car: it might be ₹100 for the first hour, and then ₹200 for anything over one hour up to two hours. At exactly one hour, the cost "jumps."
* A Hole: There's a point missing from the graph. Maybe the function is defined for all values except one, creating an empty circle on the graph. For instance, consider f(x) = (x² - 4)/(x - 2). At x=2, the function is undefined, creating a hole.
* A Vertical Asymptote (Break): The function shoots off to positive or negative infinity at a certain point. Think of f(x) = 1/x at x=0. The graph breaks apart, one piece going up, the other going down.

If a function has none of these issues at a particular point, it's considered continuous at that point. If it has any of these issues, it's discontinuous.

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### 2. The Formal Definition of Continuity at a Point

While the graphical intuition is great for understanding, mathematics demands precision. For a function $f(x)$ to be continuous at a specific point $x = a$, three conditions must all be met. If even one condition fails, the function is discontinuous at that point.

The conditions for $f(x)$ to be continuous at $x=a$ are:

1. $f(a)$ must be defined.
* This means that the value of the function at $x=a$ must exist and be a finite real number. If $f(a)$ is undefined (e.g., division by zero, square root of a negative number, logarithm of zero or a negative number), then the function cannot be continuous at $x=a$. Graphically, this corresponds to a "hole" or a "break" where the function value itself is missing.

2. $lim_{x o a} f(x)$ must exist.
* For the limit to exist at $x=a$, the Left Hand Limit (LHL) and the Right Hand Limit (RHL) must both exist and be equal.
* LHL: $lim_{x o a^-} f(x)$ (approaching $a$ from values less than $a$)
* RHL: $lim_{x o a^+} f(x)$ (approaching $a$ from values greater than $a$)
* So, $lim_{x o a^-} f(x) = lim_{x o a^+} f(x) = L$ (where L is a finite real number).
* If the LHL and RHL are not equal, or if either approaches $pm infty$, then the limit does not exist, and the function is discontinuous. Graphically, this corresponds to a "jump" or a "vertical asymptote."

3. $lim_{x o a} f(x) = f(a)$
* This is the crucial link! It means that the value the function is *approaching* as $x$ gets closer to $a$ must be exactly equal to the *actual value* of the function at $x=a$.
* If the limit exists but is not equal to $f(a)$ (e.g., there's a hole at $f(a)$ but the function is defined somewhere else at $a$), or if $f(a)$ is undefined while the limit exists, then this condition fails.

JEE Focus: It's absolutely vital to check all three conditions. Many JEE problems are designed to trick you by satisfying two conditions but failing the third. Always start by evaluating $f(a)$, then the limits, and finally compare them.

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### 3. Delving into Types of Discontinuity

When a function is not continuous at a point, we say it's discontinuous. There are different ways a function can be discontinuous, and classifying them helps in understanding their behavior and, sometimes, in "fixing" them.

#### 3.1 Removable Discontinuity

This type of discontinuity occurs when the limit of the function at a point *exists* but is not equal to the function's value at that point, or the function's value is simply *undefined* at that point. It's called "removable" because we can often redefine the function at that single point to make it continuous.

There are two sub-types:

1. Missing Point Discontinuity:
* Conditions: $lim_{x o a} f(x) = L$ (exists and is finite), but $f(a)$ is undefined.
* Graphical Intuition: A "hole" in the graph at $x=a$, where the function literally doesn't have a value.
* How to "Remove" It: Define $f(a) = L$.
* Example: $f(x) = frac{x^2 - 4}{x - 2}$.
* At $x=2$, $f(2)$ is undefined (0/0 form).
* $lim_{x o 2} frac{x^2 - 4}{x - 2} = lim_{x o 2} frac{(x-2)(x+2)}{x-2} = lim_{x o 2} (x+2) = 4$.
* The limit exists, but $f(2)$ doesn't. So, it's a removable discontinuity. We can redefine $f(x)$ as:
$g(x) = egin{cases} frac{x^2 - 4}{x - 2} & ext{if } x
eq 2 \ 4 & ext{if } x = 2 end{cases}$
Now, $g(x)$ is continuous at $x=2$.

2. Isolated Point Discontinuity:
* Conditions: $lim_{x o a} f(x) = L$ (exists and is finite), and $f(a)$ is defined, but $f(a)
eq L$.
* Graphical Intuition: A "hole" in the graph at the expected limit value, but the point $(a, f(a))$ is floating somewhere else.
* How to "Remove" It: Redefine $f(a) = L$.
* Example: $f(x) = egin{cases} x^2 & ext{if } x
eq 1 \ 5 & ext{if } x = 1 end{cases}$
* At $x=1$, $f(1) = 5$.
* $lim_{x o 1} f(x) = lim_{x o 1} x^2 = 1^2 = 1$.
* Here, $f(1) = 5
eq lim_{x o 1} f(x) = 1$. So, it's an isolated point discontinuity. We can redefine $f(1)=1$ to make it continuous.

#### 3.2 Non-Removable Discontinuity

These are more severe types of discontinuities where the limit either does not exist (LHL $
eq$ RHL) or approaches infinity. You cannot simply redefine the function at a single point to make it continuous.

1. Jump Discontinuity (or Finite Discontinuity):
* Conditions: Both LHL and RHL exist and are finite, but they are not equal ($lim_{x o a^-} f(x)
eq lim_{x o a^+} f(x)$).
* Graphical Intuition: The graph "jumps" from one y-value to another at $x=a$.
* Example: $f(x) = [x]$ (Greatest Integer Function) at $x=1$.
* $f(1) = [1] = 1$.
* $lim_{x o 1^-} [x] = lim_{h o 0} [1-h] = 0$.
* $lim_{x o 1^+} [x] = lim_{h o 0} [1+h] = 1$.
* Since $0
eq 1$, the LHL $
eq$ RHL. The limit does not exist. Hence, it's a jump discontinuity.

2. Infinite Discontinuity:
* Conditions: At least one of the one-sided limits ($lim_{x o a^-} f(x)$ or $lim_{x o a^+} f(x)$) approaches $pm infty$.
* Graphical Intuition: A vertical asymptote where the function shoots off to infinity.
* Example: $f(x) = frac{1}{x-1}$ at $x=1$.
* $f(1)$ is undefined.
* $lim_{x o 1^-} frac{1}{x-1} = -infty$.
* $lim_{x o 1^+} frac{1}{x-1} = +infty$.
* Since the limits are infinite, it's an infinite discontinuity.

3. Oscillatory Discontinuity:
* Conditions: The limit does not exist because the function oscillates infinitely many times between two fixed values as $x$ approaches $a$. The LHL and RHL do not settle on specific values.
* Graphical Intuition: The graph wiggles wildly and never settles near a specific point.
* Example: $f(x) = sin(frac{1}{x})$ at $x=0$.
* As $x o 0$, $1/x$ approaches $pm infty$. The sine function oscillates between -1 and 1 infinitely often as its argument approaches infinity. Thus, $lim_{x o 0} sin(frac{1}{x})$ does not exist. This is an oscillatory discontinuity.

JEE Focus: Being able to identify the type of discontinuity is often a part of JEE questions. For example, you might be asked to classify the discontinuities of a given function.

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### 4. Continuity Over an Interval

So far, we've discussed continuity at a single point. But what about a range of points?

1. Continuity on an Open Interval $(a, b)$:
* A function $f(x)$ is said to be continuous on an open interval $(a, b)$ if it is continuous at every single point within that interval.

2. Continuity on a Closed Interval $[a, b]$:
* This is slightly more involved because we need to consider the endpoints.
* A function $f(x)$ is continuous on a closed interval $[a, b]$ if:
1. It is continuous on the open interval $(a, b)$.
2. It is continuous from the right at $x=a$: $lim_{x o a^+} f(x) = f(a)$.
3. It is continuous from the left at $x=b$: $lim_{x o b^-} f(x) = f(b)$.

JEE Focus: Continuity on a closed interval is crucial for theorems like the Intermediate Value Theorem (IVT) and the Extreme Value Theorem (EVT), which are frequently tested in advanced problems.

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### 5. Algebra of Continuous Functions (Properties)

These properties are extremely useful for determining the continuity of complex functions without checking every single point individually.

If $f(x)$ and $g(x)$ are two functions that are continuous at $x=a$, then:

1. Sum/Difference: $f(x) pm g(x)$ is continuous at $x=a$.
2. Product: $f(x) cdot g(x)$ is continuous at $x=a$.
3. Scalar Multiple: $c cdot f(x)$ is continuous at $x=a$, where $c$ is any real constant.
4. Quotient: $frac{f(x)}{g(x)}$ is continuous at $x=a$, **provided $g(a)
eq 0$**. (If $g(a)=0$, there might be a discontinuity).
5. Composition of Functions: If $g(x)$ is continuous at $x=a$, and $f(y)$ is continuous at $y=g(a)$, then the composite function $(f circ g)(x) = f(g(x))$ is continuous at $x=a$.
* Example: If $g(x) = x^2$ (continuous everywhere) and $f(x) = sin x$ (continuous everywhere), then $f(g(x)) = sin(x^2)$ is continuous everywhere.

JEE Focus: These properties are powerful shortcuts. For instance, if you have a function like $h(x) = sin(x^2 + e^x)$, you can quickly deduce its continuity by knowing that $x^2$, $e^x$, their sum $(x^2+e^x)$, and $sin( ext{continuous function})$ are all continuous.

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### 6. Important Standard Continuous Functions

It's helpful to remember that many common functions are inherently continuous in their domains:

* Polynomial Functions: $P(x) = a_n x^n + a_{n-1} x^{n-1} + dots + a_0$ are continuous everywhere (for all $x in mathbb{R}$).
* Exponential Functions: $e^x$, $a^x$ (where $a > 0$) are continuous everywhere.
* Logarithmic Functions: $log_a x$ (where $a > 0, a
eq 1$) are continuous in their domain, i.e., for all $x > 0$.
* Trigonometric Functions:
* $sin x$, $cos x$ are continuous everywhere.
* $ an x$, $sec x$ are continuous in their domains (i.e., not at $x = (2n+1)pi/2$, where $n in mathbb{Z}$).
* $cot x$, $csc x$ are continuous in their domains (i.e., not at $x = npi$, where $n in mathbb{Z}$).
* Absolute Value Function: $|x|$ is continuous everywhere.

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### 7. Solved Examples (JEE Style)

Let's put our knowledge to the test with some examples.

Example 1: Finding a Parameter for Continuity

Question: Find the value of $k$ for which the function $f(x)$ is continuous at $x=0$:
$f(x) = egin{cases} frac{1 - cos 4x}{8x^2} & ext{if } x
eq 0 \ k & ext{if } x = 0 end{cases}$

Solution:
For $f(x)$ to be continuous at $x=0$, we need $lim_{x o 0} f(x) = f(0)$.

1. Find $f(0)$: From the definition, $f(0) = k$.

2. Find $lim_{x o 0} f(x)$: We use the first part of the function definition:
$lim_{x o 0} frac{1 - cos 4x}{8x^2}$
We know the standard limit $lim_{ heta o 0} frac{1 - cos heta}{ heta^2} = frac{1}{2}$.
Here, $ heta = 4x$. So we need $(4x)^2$ in the denominator.
$lim_{x o 0} frac{1 - cos 4x}{8x^2} = lim_{x o 0} frac{1 - cos 4x}{2 cdot (4x)^2} = frac{1}{2} lim_{x o 0} frac{1 - cos 4x}{(4x)^2}$
Applying the standard limit, we get:
$= frac{1}{2} cdot frac{1}{2} = frac{1}{4}$.

3. Equate the limit and function value:
For continuity, $lim_{x o 0} f(x) = f(0)$.
So, $frac{1}{4} = k$.

Thus, for $f(x)$ to be continuous at $x=0$, $k$ must be $frac{1}{4}$.

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Example 2: Analyzing Discontinuity of a Piecewise Function

Question: Discuss the continuity of the function $f(x) = |x-1| + [x]$ at $x=1$.

Solution:
We need to check the three conditions for continuity at $x=1$.

1. Find $f(1)$:
$f(1) = |1-1| + [1] = |0| + 1 = 0 + 1 = 1$.
So, $f(1)$ is defined and equals 1.

2. Find $lim_{x o 1} f(x)$: We need to check LHL and RHL because of $|x-1|$ and $[x]$.

* LHL: $lim_{x o 1^-} (|x-1| + [x])$
Let $x = 1 - h$, where $h o 0^+$.
$lim_{h o 0^+} (| (1-h) - 1 | + [1-h]) = lim_{h o 0^+} (| -h | + [1-h])$
$= lim_{h o 0^+} (h + 0)$ (Since $h>0$, $|-h|=h$. And for small $h>0$, $1-h$ is slightly less than 1, so $[1-h]=0$).
$= 0$.

* RHL: $lim_{x o 1^+} (|x-1| + [x])$
Let $x = 1 + h$, where $h o 0^+$.
$lim_{h o 0^+} (| (1+h) - 1 | + [1+h]) = lim_{h o 0^+} (| h | + [1+h])$
$= lim_{h o 0^+} (h + 1)$ (Since $h>0$, $|h|=h$. And for small $h>0$, $1+h$ is slightly greater than 1, so $[1+h]=1$).
$= 0 + 1 = 1$.

Since LHL ($0$) $
eq$ RHL ($1$), the limit $lim_{x o 1} f(x)$ does not exist.

3. Conclusion:
Since the limit does not exist, the function $f(x)$ is discontinuous at $x=1$.
Specifically, because both LHL and RHL existed and were finite but unequal, this is a jump discontinuity.

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### 8. CBSE vs. JEE Focus: A Comparison

Understanding the difference in focus is key to effective preparation.

Feature	CBSE (Class XII) Approach	IIT-JEE (Mains & Advanced) Approach
Definition Usage	Primarily direct application of the three conditions for continuity at a point.	In-depth application of the three conditions, often requiring advanced limit calculations (L'Hopital's Rule, series expansions, special limits).
Function Types	Mostly polynomial, simple trigonometric, exponential, and logarithmic functions; basic piecewise functions.	Complex piecewise functions, functions involving Greatest Integer Function, Fractional Part Function, Signum Function, Absolute Value, composite functions.
Discontinuity Types	Implicit understanding of discontinuity (function fails one of the conditions). Explicit classification of types is less emphasized.	Explicit identification and classification of removable, jump, infinite, and oscillatory discontinuities are often tested.
Properties of Continuous Functions	Basic understanding of sum, difference, product, quotient, and simple composition.	Strong emphasis on applying algebra of continuous functions, and particularly the continuity of composite functions, to analyze complex expressions. Often involves multiple nested functions.
Interval Continuity	Basic checks at endpoints.	Rigorous application, especially for theorems like IVT (Intermediate Value Theorem) and EVT (Extreme Value Theorem) for existence proofs or range determination.
Problem Complexity	Direct, computational problems. Finding unknown constants to ensure continuity.	Conceptual, multi-step problems. Involves finding parameters, identifying points/intervals of discontinuity, or proving continuity/discontinuity using advanced limit techniques. Often combined with differentiability.
Graphical Interpretation	Helpful for understanding.	Essential for visualizing and quickly inferring behavior, especially for functions like [x], |x|, sgn(x).

Continuity is not just a stepping stone to differentiability; it's a powerful concept in its own right. A solid grasp of continuity will make your journey through calculus much smoother and prepare you for the intricate problems you'll encounter in JEE. Keep practicing, and always remember to check those three crucial conditions!

Understanding and quickly recalling the conditions for continuity is crucial for both board exams and JEE. Here are some effective mnemonics and shortcuts to help you remember the core concepts:

1. Condition for Continuity at a Point (JEE & CBSE)

For a function f(x) to be continuous at x = a, three conditions must be met:

• f(a) must exist (the function is defined at that point).


• lim (x→a-) f(x) must exist (Left Hand Limit).


• lim (x→a+) f(x) must exist (Right Hand Limit).


• And all three must be equal: lim (x→a-) f(x) = lim (x→a+) f(x) = f(a).

Mnemonic: "LRA = Defined"

• L: Left Hand Limit exists.


• R: Right Hand Limit exists.


• A: Value of the function at 'a' (f(a)) exists.


• Defined: All three (L, R, A) are equal and a finite value.

This simple mnemonic helps you quickly recall the full set of conditions required for continuity at a specific point.

2. Types of Discontinuity (JEE Focus)

Discontinuities are broadly classified into two types: Removable and Non-Removable. Non-Removable discontinuities can be further categorised.

Mnemonic: "R.I.J.O. Discontinuities"

This helps you remember the main types of discontinuities:

• R: Removable Discontinuity (JEE: Point discontinuity)
  
  
  
  • Shortcut: Looks like a "hole" in the graph. LHL = RHL, but either f(a) is undefined or f(a) ≠ LHL = RHL. You can "redefine" f(a) to make it continuous.
  
  
  
  


• I: Infinite Discontinuity
  
  
  
  • Shortcut: Function shoots off to `±∞` near `x=a`. This often involves vertical asymptotes. Limit does not exist.
  
  
  
  


• J: Jump Discontinuity (JEE: Discontinuity of the first kind)
  
  
  
  • Shortcut: The graph "jumps" from one value to another. LHL ≠ RHL, but both exist and are finite.
  
  
  
  


• O: Oscillatory Discontinuity (JEE: Discontinuity of the second kind)
  
  
  
  • Shortcut: Function oscillates too wildly near `x=a` for the limit to exist. (e.g., `sin(1/x)` at `x=0`).
  
  
  
  

3. Algebra of Continuous Functions (JEE & CBSE)

This shortcut helps you determine the continuity of functions formed by combining continuous functions.

Mnemonic: "Basic Ops Preserve Continuity (with a catch)"

If functions f(x) and g(x) are continuous at x=a, then:

• Sum/Difference: f(x) ± g(x) is continuous at x=a.


• Product: f(x) * g(x) is continuous at x=a.


• Scalar Multiple: c * f(x) is continuous at x=a (where `c` is a constant).


• Quotient: f(x) / g(x) is continuous at x=a, provided `g(a) ≠ 0`. This is the "catch" you must always remember.


• Composition: If `g` is continuous at `a` and `f` is continuous at `g(a)`, then `f(g(x))` is continuous at `a`.

Shortcut: Generally, basic arithmetic operations maintain continuity. The only major point of failure is division by zero or when a composite function involves a discontinuity in the inner or outer function's domain.

4. Standard Continuous Functions (JEE & CBSE)

Many common functions are continuous on their respective domains. Knowing these saves time in problem-solving as you don't need to prove their continuity every time.

Mnemonic: "P.T.E.L.A. - Always C (on domain)"

This helps you recall the common families of functions that are always continuous:

• P: Polynomial Functions (e.g., `x^2 - 3x + 5`) - Continuous everywhere.


• T: Trigonometric Functions (`sin x`, `cos x`, `tan x`, etc.) - Continuous on their respective domains. (e.g., `tan x` is discontinuous at `x = (2n+1)π/2`).


• E: Exponential Functions (`e^x`, `a^x`) - Continuous everywhere.


• L: Logarithmic Functions (`log x`) - Continuous on their respective domains (for `x > 0`).


• A: Absolute Value Functions (`|x|`) - Continuous everywhere.

5. Intermediate Value Theorem (IVT) (JEE Focus)

This theorem is crucial for proving the existence of roots or specific function values.

Mnemonic: "IVT: Continuous Path, All Values In-Between"

• If `f(x)` is continuous on a closed interval `[a, b]`, AND


• `f(a) ≠ f(b)` (the function takes different values at the endpoints), THEN


• For every value `k` that lies strictly between `f(a)` and `f(b)`, there exists at least one `c` in the open interval `(a, b)` such that `f(c) = k`.

Shortcut: Think of drawing a line without lifting your pen. If you start at one height and end at another, you must have passed through all the heights in between. The "continuous path" is the key condition.

Mastering these mnemonics and shortcuts will significantly boost your speed and accuracy in solving continuity problems in exams. Keep practicing!

Quick Tips for Continuity

Continuity is a fundamental concept in calculus, frequently tested in JEE Main. Mastering it requires a clear understanding of definitions and smart application to various function types. Here are some quick tips to ace questions on continuity:

• 
  The Three Conditions of Continuity: For a function $f(x)$ to be continuous at a point $x=a$, all three conditions must be met:
  
  
  
  1. $f(a)$ must be defined. (The function value exists at $x=a$)
  
  
  2. $lim_{x o a} f(x)$ must exist. (The left-hand limit (LHL) equals the right-hand limit (RHL) at $x=a$)
  
  
  3. $lim_{x o a} f(x) = f(a)$. (The limit value equals the function value)
  
  
  
  If any of these conditions fail, the function is discontinuous at $x=a$.
  


• 
  Master Piecewise Functions: This is a JEE Main favorite. For piecewise-defined functions, always check continuity at the 'breaking points' (the points where the definition of the function changes) using the LHL, RHL, and $f(a)$ approach. Also, ensure each piece is continuous within its respective interval.
  


• 
  Common Continuous Functions (Unless Otherwise Specified):
  
  
  
  • Polynomials: Always continuous for all real numbers.
  
  
  • Exponential functions ($a^x, e^x$): Always continuous for all real numbers.
  
  
  • Trigonometric functions ($sin x, cos x$): Always continuous for all real numbers. ($ an x, cot x, sec x, csc x$ are continuous in their respective domains).
  
  
  • Logarithmic functions ($log_a x$): Continuous for $x>0$.
  
  
  • Absolute value function ($|x|$): Continuous for all real numbers.
  
  
  
  


• 
  Rational Functions: A rational function $frac{P(x)}{Q(x)}$ (where $P(x)$ and $Q(x)$ are polynomials) is continuous everywhere except at points where the denominator $Q(x)=0$. These are points of discontinuity.
  


• 
  Greatest Integer Function (GIF) and Fractional Part Function (FPF):
  
  
  
  • $f(x) = [x]$ (Greatest Integer Function) is discontinuous at all integer points ($x in mathbb{Z}$).
  
  
  • $f(x) = {x}$ (Fractional Part Function) is also discontinuous at all integer points ($x in mathbb{Z}$).
  
  
  
  Be very careful when these functions are part of a composite function.
  


• 
  Properties of Continuous Functions:
  If $f(x)$ and $g(x)$ are continuous at $x=a$, then:
  
  
  
  • $f(x) pm g(x)$ is continuous at $x=a$.
  
  
  • $f(x) cdot g(x)$ is continuous at $x=a$.
  
  
  • $frac{f(x)}{g(x)}$ is continuous at $x=a$, provided $g(a)
    eq 0$.
  
  
  • $f(g(x))$ (composite function) is continuous at $x=a$ if $g(x)$ is continuous at $x=a$ and $f(x)$ is continuous at $g(a)$.
  
  
  
  


• 
  Continuity in an Interval:
  
  
  
  • Open Interval $(a,b)$: $f(x)$ is continuous on $(a,b)$ if it is continuous at every point in the interval.
  
  
  • Closed Interval $[a,b]$ (CBSE & JEE): $f(x)$ is continuous on $[a,b]$ if:
    
    
    
    1. It is continuous on the open interval $(a,b)$.
    
    
    2. $lim_{x o a^+} f(x) = f(a)$ (Right continuous at $a$).
    
    
    3. $lim_{x o b^-} f(x) = f(b)$ (Left continuous at $b$).
    
    
    
    
  
  
  
  


• 
  Differentiability Implies Continuity: If a function is differentiable at a point, it must be continuous at that point. However, the converse is not true (e.g., $f(x)=|x|$ is continuous at $x=0$ but not differentiable at $x=0$). This can be a useful shortcut: if you've already shown differentiability, continuity is a given.
  


• 
  Removable Discontinuity: If $lim_{x o a} f(x)$ exists but $f(a)$ is either undefined or $f(a)
  eq lim_{x o a} f(x)$, the discontinuity is removable. You can make the function continuous by redefining $f(a)$ to be equal to $lim_{x o a} f(x)$. This is often tested by asking for the value of a constant that makes the function continuous.
  

Intuitive Understanding of Continuity

Continuity, at its core, describes a property of a function where its graph can be drawn without lifting your pen from the paper. Imagine sketching a function's curve on a piece of paper. If you can complete the entire drawing from one end to the other without having to pick up your pen, the function is continuous over that interval. This "no breaks, no jumps, no holes" idea forms the fundamental intuitive understanding of continuity.

When a function is continuous at a particular point, it means that the function's value at that point is exactly what you would "expect" it to be based on the values of the function in its immediate vicinity. There are no surprises, sudden shifts, or missing points.

What Makes a Function Discontinuous?

Discontinuities occur when the "no lift of pen" rule is violated. There are three primary types of breaks you'll encounter, each challenging the intuitive idea of smoothness:

1. Holes (Removable Discontinuities):
* This occurs when there's a single point missing from the graph, or a single point defined at a different location, but the graph on either side of that point otherwise connects smoothly.
* Think of it like a tiny pinprick in an otherwise perfectly drawn line. If you were to "fill in" that single point, the function would become continuous.

2. Jumps (Non-Removable Discontinuities):
* This happens when the function "jumps" from one value to another at a specific point. The graph approaches different values from the left and right sides of the point.
* Imagine drawing a line, then lifting your pen, moving it up or down, and starting a new line from a different vertical position at the same x-coordinate. Step functions are classic examples of this.

3. Vertical Asymptotes (Non-Removable, Infinite Discontinuities):
* Here, the function's value shoots off to positive or negative infinity as it approaches a certain x-value. The graph becomes infinitely tall or deep.
* You cannot connect the graph across an asymptote because the function's value becomes unbounded.

Why is Intuitive Understanding Important?

For both CBSE Board Exams and JEE Main, this intuitive understanding serves as an excellent starting point. It helps you quickly identify potential points of discontinuity by visually inspecting a graph or considering the nature of the function (e.g., division by zero, piecewise definitions). However, it is crucial to remember that while intuition guides, formal definitions and rigorous proofs are required for precise problem-solving, especially in competitive exams like JEE.

A strong intuitive grasp of continuity also helps in understanding related concepts, such as differentiability. A function must first be continuous at a point to even be considered differentiable at that point. If you can't draw it without lifting your pen, you certainly can't draw a unique tangent at that point!

"Continuity is the backbone of calculus; understand it deeply, and the path ahead becomes clearer."

While mastering the mathematical definition and problem-solving techniques for continuity is crucial for JEE and board exams, understanding its real-world applications provides a deeper appreciation for this fundamental concept.

Continuity, in essence, describes processes or quantities that change smoothly without abrupt jumps, breaks, or instantaneous teleportations. This characteristic is prevalent in many natural phenomena and engineered systems:

• 
  Physics and Engineering (Motion & Physical Properties):
  
  
  
  • 
    Position and Velocity: When a car moves from one point to another, its position is a continuous function of time. It cannot instantaneously disappear from one spot and reappear at another; it must traverse all intermediate points. Similarly, its velocity, while it can change rapidly (acceleration), typically changes continuously.
    
  
  
  • 
    Temperature, Pressure, and Density: In a given medium, quantities like temperature, pressure, or density usually vary continuously from one point to another. There aren't sudden, infinite jumps in temperature across an infinitesimal distance unless there's a phase transition boundary (which can be modeled as a discontinuity).
    
  
  
  • 
    Structural Integrity: In designing bridges or buildings, engineers assume that stress and strain within a material are continuous functions. A sudden, unpredicted discontinuity in stress could indicate a point of failure.
    
  
  
  
  


• 
  Economics and Finance:
  
  
  
  • 
    Supply and Demand Curves: In economic models, supply and demand are often represented by continuous functions. This implies that small changes in price lead to small, predictable changes in the quantity supplied or demanded, rather than sudden, unpredictable shifts.
    
  
  
  • 
    Stock Prices and Interest Rates: While often observed at discrete intervals, financial models frequently treat asset prices and interest rates as continuously evolving processes (e.g., in continuous-time financial mathematics) to capture market dynamics more accurately.
    
  
  
  
  


• 
  Computer Graphics and Animation:
  
  
  
  • 
    To create smooth, realistic animations and visual effects, the functions describing the position, rotation, and scaling of objects over time must be continuous. Discontinuities would result in jerky movements, objects "teleporting," or sudden changes that break the illusion of reality.
    
  
  
  
  


• 
  Biology and Medicine:
  
  
  
  • 
    Drug Concentration: The concentration of a drug in a patient's bloodstream typically changes continuously over time after administration, rising to a peak and then gradually decaying. This continuous model helps determine dosage and timing.
    
  
  
  • 
    Growth Curves: Biological growth processes, such as the growth of a population or the height of an organism, are often modeled as continuous functions over time.
    
  
  
  
  

JEE & CBSE Perspective: While these applications help build intuition, remember that exams like JEE Main and CBSE boards primarily focus on your ability to apply the formal definition of continuity (using limits) to various functions, identify points of discontinuity, and solve related problems. Conceptual understanding from real-world examples can sometimes help in visualizing function behavior, but direct application questions are rare.

Understanding abstract mathematical concepts like continuity can be greatly aided by drawing parallels to everyday experiences. Analogies help bridge the gap between abstract definitions and intuitive understanding. Here are some common analogies for continuity:

---

1. Drawing a Curve Without Lifting Your Pen

• 
  The Analogy: Imagine you are drawing the graph of a function on a piece of paper. If you can draw the entire graph within a certain interval without lifting your pen from the paper, then the function is considered continuous over that interval.
  


• 
  Mathematical Connection:
  
  
  
  • 
    No Lifts: This directly corresponds to the idea that there are no breaks, gaps, or holes in the function's graph.
    
  
  
  • 
    Smooth Motion: This implies that as you move along the x-axis, the y-value changes smoothly without any sudden jumps or drops.
    
  
  
  • 
    At a Point: For a function to be continuous at a specific point 'c', it means your pen would pass through that point without any interruption. This reflects the condition $lim_{x o c^-} f(x) = lim_{x o c^+} f(x) = f(c)$.
    
  
  
  
  


• 
  Discontinuity Analogy: If you have to lift your pen to continue drawing (e.g., to jump to another part of the graph, or because there's a hole), that signifies a point of discontinuity.
  

---

2. A Smooth Road or Bridge

• 
  The Analogy: Consider a road you are driving on. If the road is continuous, you can drive from one point to another without any obstruction, sudden drop-offs, or missing sections.
  


• 
  Mathematical Connection:
  
  
  
  • 
    Uninterrupted Travel: This represents the path of the function. For every input (x-value), there's a corresponding, well-defined output (y-value), and you can smoothly transition between points.
    
  
  
  • 
    No Broken Bridges/Missing Sections: These would be points of discontinuity.
    
    
    
    • A hole in the road (a point where the function is undefined but limits exist) means you can see where the road *should* go, but you can't actually drive through that exact spot.
    
    
    • A broken bridge or a sudden cliff (a jump discontinuity) means the road abruptly ends and restarts at a different level, making travel impossible across that point.
    
    
    • A bottomless pit (an infinite discontinuity) means the road plunges infinitely downwards or upwards, making passage impossible.
    
    
    
    
  
  
  
  


• 
  JEE Relevance: This analogy is particularly useful for visualizing different types of discontinuities (removable, jump, infinite) and understanding why specific points break the "smoothness" of the path.
  

---

3. A Chain or String

• 
  The Analogy: Imagine a physical chain or a piece of string. If it's continuous, all its links are connected, or the string has no cuts. It forms one unbroken piece.
  


• 
  Mathematical Connection:
  
  
  
  • 
    Connected Links/Uncut String: This represents the connected nature of a continuous function's domain and range. Each point is "linked" to its neighbors.
    
  
  
  • 
    Broken Link/Cut String: A broken link in the chain or a cut in the string signifies a point of discontinuity. The chain is no longer one single, continuous entity. This helps emphasize that a function's "value" must be present and correctly aligned with its "approaching values."
    
  
  
  
  

By relating these everyday scenarios to the formal definition of continuity, you can develop a stronger intuition for this crucial calculus concept. Remember, a function is continuous if its graph is one unbroken curve without any holes, jumps, or asymptotes.

Prerequisites for Continuity

Understanding the concept of continuity of a function is fundamental in Calculus and forms a cornerstone for differentiability. Before diving into continuity, a solid grasp of certain foundational topics is essential. Skipping these prerequisites often leads to misconceptions and difficulty in solving continuity problems effectively.

Here are the key concepts you must be comfortable with:

• 
  1. Functions:
  
  
  
  • Definition and Types: A clear understanding of what a function is, its domain, and range. Familiarity with various types of functions such as polynomial, rational, trigonometric, exponential, logarithmic, and especially piecewise functions, is crucial.
  
  
  • Graphs of Functions: The ability to sketch the graphs of standard functions and understand their behavior helps immensely in visualizing continuity. Many continuity problems can be quickly analyzed by looking at the graph.
  
  
  
  


• 
  2. Limits:
  
  
  
  • This is the MOST CRITICAL prerequisite. Continuity is formally defined using limits. Without a strong understanding of limits, continuity cannot be grasped.
  
  
  • Concept of a Limit: What it means for a function to approach a certain value as the input approaches a specific point. Understanding Left Hand Limit (LHL) and Right Hand Limit (RHL) is paramount, as continuity at a point explicitly involves comparing LHL, RHL, and the function's value.
  
  
  • Evaluation of Limits: Proficiency in evaluating limits using various techniques is necessary. This includes:
    
    
    
    • Direct substitution (if determinate form).
    
    
    • Factorization and rationalization for indeterminate forms ($0/0$).
    
    
    • Using standard limits (e.g., $lim_{x o 0} frac{sin x}{x} = 1$).
    
    
    • L'Hôpital's Rule (especially important for JEE Main and Advanced, less emphasized in CBSE for the direct definition of limit evaluation but useful for indeterminate forms).
    
    
    • Limits involving exponential and logarithmic functions.
    
    
    
    
  
  
  • Limits at Infinity: Understanding the behavior of functions as $x o infty$ or $x o -infty$.
  
  
  
  


• 
  3. Algebraic Manipulations and Inequalities:
  
  
  
  • Algebra: Basic algebraic skills like factoring, solving equations, and simplifying expressions are frequently used when evaluating limits and defining function values.
  
  
  • Inequalities: The ability to solve inequalities is important for determining domains of functions, intervals of continuity, or when dealing with conditions for continuity.
  
  
  
  


• 
  4. Trigonometry:
  
  
  
  • Knowledge of fundamental trigonometric identities and properties of trigonometric functions is necessary, especially when dealing with trigonometric functions within continuity problems.
  
  
  
  

A strong foundation in these areas will make your journey through Continuity much smoother and more productive. Ensure you revisit these topics if you find yourself struggling with continuity concepts.

Related Topics to Continuity

Understanding continuity is foundational in calculus and is deeply interconnected with several other key mathematical concepts. A solid grasp of these related topics will significantly enhance your ability to solve problems involving functions and their behavior. Here are the core topics that are either prerequisites for, extensions of, or direct applications of continuity:

• 
  

  Limits

  
  

  Limits are the bedrock of continuity. The formal definition of continuity at a point relies entirely on the concept of limits. A function ( f(x) ) is continuous at a point ( x=a ) if and only if the limit of ( f(x) ) as ( x ) approaches ( a ) exists and is equal to the function's value at ( a ).

  
  
  
  
  • Mathematically: ( lim_{x o a} f(x) = f(a) ).
  
  
  • This implies that the Left Hand Limit (LHL), Right Hand Limit (RHL), and the function value at ( a ) must all be equal: ( lim_{x o a^-} f(x) = lim_{x o a^+} f(x) = f(a) ).
  
  
  • JEE Relevance: Almost every problem on continuity requires a strong command over evaluating limits, including indeterminate forms and standard limits.
  
  
  
  


• 
  

  Differentiability

  
  

  There's a crucial relationship between differentiability and continuity: If a function is differentiable at a point, then it must be continuous at that point. However, the converse is not true; a function can be continuous but not differentiable (e.g., ( f(x) = |x| ) at ( x=0 )).

  
  
  
  
  • JEE Relevance: This relationship is frequently tested in both conceptual questions and multi-part problems. Recognizing that differentiability implies continuity can simplify problem-solving steps.
  
  
  
  


• 
  

  Types of Discontinuities

  
  

  To fully understand continuity, one must also understand what it means for a function to be discontinuous. Classifying discontinuities (e.g., removable, jump, infinite discontinuity) helps in analyzing the behavior of functions that are not continuous over their entire domain.

  
  
  
  
  • JEE Relevance: Problems often ask to identify points of discontinuity or determine parameters that make a function continuous, which implicitly involves understanding different types of discontinuities.
  
  
  
  


• 
  

  Intermediate Value Theorem (IVT)

  
  

  The IVT is a powerful theorem that directly applies to continuous functions. It states that if a function ( f ) is continuous on a closed interval ( [a, b] ), and ( k ) is any value between ( f(a) ) and ( f(b) ), then there must exist at least one point ( c ) in the open interval ( (a, b) ) such that ( f(c) = k ).

  
  
  
  
  • JEE Relevance: Used extensively in proving the existence of roots for equations (where ( k=0 )) and for understanding the range of continuous functions over an interval.
  
  
  
  


• 
  

  Extreme Value Theorem (EVT)

  
  

  Another fundamental theorem for continuous functions. The EVT states that if a function ( f ) is continuous on a closed interval ( [a, b] ), then ( f ) must attain both an absolute maximum and an absolute minimum value on that interval.

  
  
  
  
  • JEE Relevance: Essential for optimization problems, especially when finding global maximum or minimum values for functions defined on closed intervals.
  
  
  
  


• 
  

  Properties of Continuous Functions

  
  

  Understanding that sums, differences, products, quotients (where the denominator is non-zero), and compositions of continuous functions are also continuous simplifies the analysis of complex functions. For example, if ( f(x) ) and ( g(x) ) are continuous, then ( f(x) + g(x) ), ( f(x) cdot g(x) ), and ( f(g(x)) ) are also continuous (under suitable conditions).

  
  
  
  
  • JEE Relevance: These properties are implicitly used when dealing with polynomials, trigonometric functions, exponential functions, and logarithmic functions, which are all continuous in their respective domains.
  
  
  
  

Mastering these interconnected concepts will provide a holistic understanding of function behavior, which is vital for advanced topics in calculus.

Navigating continuity problems in competitive exams like JEE Main requires not just a strong theoretical understanding but also an awareness of common pitfalls. Many questions are designed to test your attention to detail and ability to spot subtle traps. Being aware of these can significantly improve your accuracy and score.

Common Exam Traps in Continuity

• 
  Trap 1: Overlooking Domain Restrictions
  
  
  Students often jump directly to checking LHL, RHL, and function value without first considering the function's domain. A function can only be continuous on its domain. If a point is not in the domain, continuity cannot be discussed there.
  
  
  Tip: Always identify the domain of the function first. Points outside the domain are not points of discontinuity; rather, the function is simply not defined there.
  
  
  Example: For $f(x) = frac{1}{x-2}$, $x=2$ is not a point of discontinuity but a point where the function is undefined. $f(x)$ is continuous on its domain, i.e., $R - {2}$.
  


• 
  Trap 2: Incomplete Analysis for Piecewise Functions
  
  
  For functions defined piecewise, students usually check continuity only at the "break points" where the definition changes. However, it's crucial to also ensure that each individual piece is continuous within its respective interval.
  
  
  Tip:
  
  
  
  1. First, check the continuity of each sub-function in its defined interval.
  
  
  2. Then, check continuity at the 'joining' points (where the definition changes) by comparing LHL, RHL, and function value.
  
  
  
  
  


• 
  Trap 3: Mismanaging Modulus and GIF/FPF Functions
  
  
  Functions involving modulus (|x|), greatest integer function ([x]), or fractional part function ({x}) are frequent sources of discontinuity.
  
  
  
  • 
    Modulus: While |f(x)| is continuous if f(x) is continuous, functions like f(x) = x/|x| or f(x) = x + |x| (at x=0) need careful LHL/RHL analysis.
    
  
  
  • 
    GIF/FPF: These functions are discontinuous at all integer points. Questions often involve compositions like [sin x] or sin([x]), where discontinuities can occur when the argument of GIF becomes an integer.
    
  
  
  
  Tip: For modulus functions, identify points where the argument of the modulus becomes zero. For GIF/FPF, identify points where the argument becomes an integer. Perform LHL/RHL analysis meticulously at these points.
  


• 
  Trap 4: Assuming Continuity for Rational Functions (Denominator Zero)
  
  
  When dealing with rational functions of the form f(x)/g(x), a common mistake is to overlook points where the denominator g(x) = 0. These points are potential discontinuities (removable or non-removable).
  
  
  Tip: Always factorize and simplify the expression if possible to identify removable discontinuities. If simplification is not possible, $g(x)=0$ leads to a non-removable discontinuity.
  
  
  Example: For $f(x) = frac{x^2-4}{x-2}$, $x=2$ is a point of removable discontinuity. For $g(x) = frac{1}{x-2}$, $x=2$ is a point of non-removable (infinite) discontinuity.
  


• 
  Trap 5: Misapplying Properties of Continuous Functions (JEE Specific)
  
  
  While the sum, difference, and product of continuous functions are continuous, and the quotient is continuous where the denominator is non-zero, traps appear in compositions or inverse functions.
  
  
  
  • 
    Composition: If g(x) is continuous at x=a and f(x) is continuous at x=g(a), then f(g(x)) is continuous at x=a. A common trap is assuming continuity when one of these conditions fails.
    
  
  
  • 
    Inverse Functions: The continuity of an inverse function f⁻¹(x) is related to the continuity and strict monotonicity of f(x). Be cautious with functions that are not strictly monotonic.
    
  
  
  
  Tip: For composite functions, always check the continuity of the inner function at the given point and the outer function at the value of the inner function at that point.
  

By being mindful of these common traps and practicing with a keen eye for detail, you can significantly improve your performance on continuity problems in both JEE Main and board examinations. Always show clear working for CBSE, and be quick and accurate for JEE.

Continuity is a fundamental concept in calculus, bridging the gap between algebraic manipulation and the behavior of functions. Mastering its principles is crucial for both Board exams and JEE Main, as it forms the basis for differentiability and many other advanced topics.

Key Takeaways for Continuity

• 
  Definition at a Point: A function $f\left(x\right)$ is continuous at a point $x=a$ if and only if all three of the following conditions are met:
  
  
  
  1. $f\left(a\right)$ exists (the function is defined at $x=a$).
  
  
  2. $\underset{x\to a}{\lim }f\left(x\right)$ exists (the left-hand limit and right-hand limit are equal and finite).
  
  
  3. $\underset{x\to a}{\lim }f\left(x\right)=f\left(a\right)$ (the limit value equals the function value).
  
  
  
  JEE Tip: For piecewise functions, always check continuity at the 'critical' points where the definition of the function changes. This is a common JEE question type.
  


• 
  Continuity in an Interval:
  
  
  
  • A function $f\left(x\right)$ is continuous in an open interval $(a,b)$ if it is continuous at every point within that interval.
  
  
  • A function $f\left(x\right)$ is continuous in a closed interval $[a,b]$ if:
    
    
    
    1. It is continuous in $(a,b)$.
    
    
    2. $\underset{x\to a^{+}}{\lim }f\left(x\right)=f\left(a\right)$ (right continuous at $a$).
    
    
    3. $\underset{x\to b^{-}}{\lim }f\left(x\right)=f\left(b\right)$ (left continuous at $b$).
    
    
    
    
  
  
  
  


• 
  Types of Discontinuities:
  
  
  
  • Removable Discontinuity: Occurs if $\underset{x\to a}{\lim }f\left(x\right)$ exists but is not equal to $f\left(a\right)$ (point discontinuity) or $f\left(a\right)$ is undefined (missing point discontinuity/hole). This can be 'removed' by redefining $f\left(a\right)$.
  
  
  • Non-Removable Discontinuity: Occurs if $\underset{x\to a}{\lim }f\left(x\right)$ does not exist.
    
    
    
    • Jump Discontinuity: LHL $\ne$ RHL (e.g., $\left[x\right]$ (Greatest Integer Function) at integer points).
    
    
    • Infinite Discontinuity: One or both limits are $\pm \infty$ (e.g., $1/x$ at $x=0$).
    
    
    • Oscillatory Discontinuity: The function oscillates rapidly without approaching a single value (e.g., $\sin (1/x)$ at $x=0$).
    
    
    
    
  
  
  
  


• 
  Continuity of Standard Functions:
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  

  Function Type	Continuity Domain
  Polynomials ($x^n$, $x^2+3x+1$)	All Real Numbers ($ℝ$)
  Exponential ($e^x$, $a^x$)	All Real Numbers ($ℝ$)
  Logarithmic ($\log x$)	Domain of the function ($x>0$)
  Trigonometric ($\sin x,\cos x$)	All Real Numbers ($ℝ$)
  Trigonometric ($\tan x,\sec x$)	Where defined ($x\ne (2n+1)\pi /2$)

  
  


• 
  Algebra of Continuous Functions: If $f$ and $g$ are continuous at $x=a$, then:
  
  
  
  • $f\pm g$ is continuous at $x=a$.
  
  
  • $f\cdot g$ is continuous at $x=a$.
  
  
  • $f/g$ is continuous at $x=a$, provided $g\left(a\right)\ne 0$.
  
  
  • $kf$ is continuous at $x=a$ for any constant $k$.
  
  
  • The composite function $g\circ f$ (i.e., $g\left(f\right(x\left)\right)$) is continuous at $x=a$ if $f$ is continuous at $a$ and $g$ is continuous at $f\left(a\right)$.
  
  
  
  


• 
  CBSE vs. JEE Focus:
  
  
  
  • CBSE: Expect questions typically involving piecewise functions, finding parameters to make a function continuous at a point, or proving continuity of standard functions using the definition. Graphical interpretation is also important.
  
  
  • JEE Main: Questions can be more complex, involving Greatest Integer Function ($\left[x\right]$), Fractional Part Function ($\left\{x\right\}$), signum function ($\mathrm{sgn}\left(x\right)$), absolute value functions, and combinations of these. The Intermediate Value Theorem (IVT) is also a concept frequently tested indirectly in JEE problems to determine roots or existence of values.
  
  
  
  

Remember, a solid understanding of limits is the prerequisite for mastering continuity. Practice problems that involve finding limits for piecewise functions and identifying types of discontinuities to strengthen your grasp.

A systematic problem-solving approach is crucial for successfully tackling continuity problems in both JEE Main and board exams. While the fundamental definition remains the same, the complexity of functions can vary significantly.

General Problem Solving Approach for Continuity at a Point

The core principle of continuity at a point 'a' is that the function must exist at 'a', and its value must match the limit as x approaches 'a'.

1. Identify the Point of Interest: First, determine the specific point 'a' at which continuity needs to be checked. This is often explicitly given or implied (e.g., points where the function definition changes for piecewise functions, or points where the denominator might be zero).


2. Evaluate the Function Value (f(a)): Calculate the exact value of the function at the point 'a'. If f(a) is undefined (e.g., due to division by zero, log of non-positive, square root of negative), the function is discontinuous at 'a'.


3. Calculate Left Hand Limit (LHL): Determine $lim_{x o a^-} f(x)$. This involves considering values of x slightly less than 'a'.


4. Calculate Right Hand Limit (RHL): Determine $lim_{x o a^+} f(x)$. This involves considering values of x slightly greater than 'a'.


5. Compare and Conclude:
   
   
   
   • If LHL ≠ RHL, then $lim_{x o a} f(x)$ does not exist, and thus the function is discontinuous at 'a'.
   
   
   • If LHL = RHL = L, then $lim_{x o a} f(x) = L$ exists.
   
   
   • Now, compare L with f(a):
     
     
     
     • If LHL = RHL = f(a), then the function is continuous at 'a'.
     
     
     • If LHL = RHL ≠ f(a), then the function is discontinuous at 'a' (removable discontinuity).
     
     
     
     
   
   
   
   

Approach for Continuity in an Interval

To check continuity over an interval:

• Open Interval (a, b): A function is continuous in an open interval (a, b) if it is continuous at every point within that interval.


• Closed Interval [a, b]: A function is continuous in a closed interval [a, b] if:
  
  
  
  • It is continuous in the open interval (a, b).
  
  
  • $lim_{x o a^+} f(x) = f(a)$ (Right continuous at 'a').
  
  
  • $lim_{x o b^-} f(x) = f(b)$ (Left continuous at 'b').
  
  
  
  

Specific Strategies for Different Function Types

Function Type	Problem Solving Strategy
Piecewise Functions	Check continuity at the "breaking points" where the function definition changes. Check continuity in each sub-interval (where the function is defined by a single expression), treating them as standard functions.
Functions involving Modulus (|x|)	Rewrite the function by removing the modulus sign based on the sign of the expression inside it. This converts it into a piecewise function. Then apply the piecewise function strategy. Critical points are where the expression inside the modulus becomes zero.
Functions involving Greatest Integer ([x]) or Fractional Part ({x})	These functions are generally discontinuous at integer points. Focus on checking continuity at integer values of the argument. For $f(x) = [g(x)]$, check where $g(x)$ is an integer.
Composite Functions (e.g., f(g(x)))	A composite function $f(g(x))$ is continuous if $g(x)$ is continuous at 'a' and $f(x)$ is continuous at $g(a)$. Identify potential points of discontinuity for both $f(x)$ and $g(x)$.

JEE Main vs. CBSE Board Exam Focus

• CBSE Boards: Problems are usually direct, often involving piecewise functions with simple algebraic or trigonometric expressions, requiring a straightforward application of the LHL=RHL=f(a) condition. Parameter finding is common.


• JEE Main: Expect more complex functions (e.g., involving mixtures of modulus, greatest integer, exponential, or inverse trigonometric functions), composite functions, or problems testing the understanding of discontinuity types. Often, you might need to find parameters that make a function continuous everywhere or over a specific interval. Identifying points of discontinuity for domain restriction is also common.

Always remember to simplify the function as much as possible before applying limits. Understanding the graphs of common functions (polynomials, trigonometric, exponential, logarithmic, modulus, GIF) can also provide quick insights into their continuity.

For students preparing for their CBSE board examinations, the topic of Continuity is fundamental and frequently tested. The questions are generally direct, focusing on the definition and its applications. Mastering these concepts ensures a strong score in this section.

Key Focus Areas for CBSE Boards:

1. 
   

   Understanding the Definition of Continuity at a Point

   
   

   This is the bedrock of all CBSE problems on continuity. A function (f(x)) is continuous at a point (x = a) if and only if:

   
   
   
   
   • (lim_{x o a^-} f(x)) (Left-Hand Limit) exists.
   
   
   • (lim_{x o a^+} f(x)) (Right-Hand Limit) exists.
   
   
   • (f(a)) (Value of the function at (x=a)) exists.
   
   
   • All three are equal:
     
     (lim_{x o a^-} f(x) = lim_{x o a^+} f(x) = f(a))
     
   
   
   
   

   Most CBSE problems revolve around verifying this condition, especially for piecewise functions.

   
   


2. 
   

   Continuity of Standard Functions

   
   

   CBSE expects you to know the continuity properties of basic functions:

   
   
   
   
   • Polynomial functions: Continuous everywhere.
   
   
   • Rational functions: Continuous in their domain (i.e., everywhere except where the denominator is zero).
   
   
   • Trigonometric functions: Continuous in their respective domains (e.g., (sin x) and (cos x) are continuous everywhere; ( an x) is continuous except where (cos x = 0)).
   
   
   • Exponential functions ((a^x)): Continuous everywhere.
   
   
   • Logarithmic functions ((log_a x)): Continuous in their domain (i.e., for (x > 0)).
   
   
   
   

   You should be able to state the domain of continuity for these functions without detailed proof.

   
   


3. 
   

   Problems Involving Piecewise Functions

   
   

   A significant portion of CBSE questions involves checking continuity or finding unknown constants for functions defined piecewise. You will typically be given a function like:

   
   

   [ f(x) = egin{cases}
   
               g(x) & 	ext{if } x < a \
   
               k & 	ext{if } x = a \
   
               h(x) & 	ext{if } x > a
   
           end{cases} ]

   
   

   To ensure continuity at (x=a), you must apply the three-part definition: calculate (lim_{x o a^-} g(x)), (lim_{x o a^+} h(x)), and compare them with (f(a) = k).

   
   


4. 
   

   Finding Unknown Constants for Continuity

   
   

   This is a very common type of question. You will be given a piecewise function, told that it is continuous at a certain point, and asked to find the value(s) of one or more constants (e.g., (k, a, b)) that make the function continuous. This requires setting the LHL, RHL, and function value equal to each other and solving the resulting equation(s).

   
   
   
   Example Type: Find the value of (k) for which the function (f(x) = egin{cases} frac{sin 2x}{x} & ext{if } x
   eq 0 \ k & ext{if } x = 0 end{cases}) is continuous at (x=0).
   
   
   


5. 
   

   Continuity in an Interval

   
   

   Understand the difference between continuity in an open interval ((a, b)) and a closed interval ([a, b]).

   
   
   
   
   • Open Interval ((a, b)): A function is continuous in ((a, b)) if it is continuous at every point in the interval.
   
   
   • Closed Interval ([a, b]): A function is continuous in ([a, b]) if it is continuous in ((a, b)), and also continuous from the right at (a) ((lim_{x o a^+} f(x) = f(a))) and continuous from the left at (b) ((lim_{x o b^-} f(x) = f(b))).
   
   
   
   


6. 
   

   Algebra of Continuous Functions

   
   

   CBSE expects knowledge of how continuity behaves under basic arithmetic operations:

   
   
   
   
   • If (f) and (g) are continuous functions, then (f pm g), (f cdot g), and (f/g) (where (g(x)
     eq 0)) are also continuous.
   
   
   • The composition of continuous functions is also continuous (i.e., if (f) and (g) are continuous, then (f circ g) is continuous).
   
   
   
   

---

CBSE vs. JEE Main Perspective:

Aspect	CBSE Board Exams	JEE Main
Complexity of Functions	Generally algebraic, trigonometric, or simple piecewise functions. Direct application of limits.	Can involve more complex functions like Greatest Integer Function ([x]), Fractional Part Function ({x}), modulus functions, and intricate algebraic manipulations.
Question Type	Mainly check continuity, find constants, or identify points of discontinuity. Step-by-step solution is expected.	Conceptual understanding, multiple conditions, sometimes involving properties of differentiable functions or complex limit evaluation. Objective type, often with options.
Focus	Accurate application of the definition and standard formulas.	Deeper conceptual understanding, problem-solving skills, and speed.

For CBSE, the emphasis is on clear, step-by-step presentation of your solutions. Ensure you write down all limit calculations and explicitly state when LHL, RHL, and (f(a)) are equal. Practice solving a variety of problems from your NCERT textbook thoroughly.

Continuity is a fundamental concept in Calculus and a frequently tested topic in JEE Main. While the basic definition is straightforward, JEE questions often involve a deeper understanding of function behavior, limits, and algebraic manipulation.

JEE Focus Areas for Continuity

Mastering continuity for JEE requires a strong grasp of the following aspects:

• Definition of Continuity at a Point:
  
  
  
  • A function f(x) is continuous at x = a if limx→a- f(x) = limx→a+ f(x) = f(a).
  
  
  • This implies that the Left Hand Limit (LHL), Right Hand Limit (RHL), and the function's value at that point must all exist and be equal.
  
  
  
  


• Continuity in an Interval:
  
  
  
  • A function is continuous in an open interval (a, b) if it is continuous at every point within that interval.
  
  
  • A function is continuous in a closed interval [a, b] if it is continuous in (a, b), and limx→a+ f(x) = f(a) and limx→b- f(x) = f(b).
  
  
  
  


• Types of Discontinuities:
  
  
  
  • Removable Discontinuity: LHL = RHL ≠ f(a) or f(a) is undefined. Can be "removed" by redefining f(a). (e.g., sin(x)/x at x=0 if f(0) is not defined as 1).
  
  
  • Non-Removable Discontinuity (Jump Discontinuity): LHL ≠ RHL. (e.g., greatest integer function, signum function).
  
  
  • Infinite Discontinuity: At least one of the limits (LHL or RHL) is ±∞. (e.g., 1/x at x=0).
  
  
  
  


• Continuity of Standard Functions:
  
  
  
  • Polynomials, exponential (a^x), logarithmic (log_ax for x>0), trigonometric (sin x, cos x) are continuous in their domains.
  
  
  • Functions involving absolute value (|f(x)|) are continuous if f(x) is continuous.
  
  
  • Special functions like Greatest Integer Function (GIF), Fractional Part Function ({x}), and Signum Function (sgn(x)) are inherently discontinuous at integer points or zero, respectively. JEE often constructs problems around these.
  
  
  
  


• Properties of Continuous Functions:
  
  
  
  • If f(x) and g(x) are continuous at x=a, then f(x) ± g(x), f(x) ⋅ g(x) are continuous at x=a.
  
  
  • f(x) / g(x) is continuous at x=a provided g(a) ≠ 0.
  
  
  • Composite Functions: If g(x) is continuous at x=a and f(x) is continuous at g(a), then f(g(x)) is continuous at x=a. This is a common trap in JEE.
  
  
  
  

Typical JEE Problem Formats

1. Finding Unknown Constants: Determine the value(s) of constants (e.g., 'a', 'b', 'k') that make a piecewise function continuous at a specific point or over an interval. This is the most common and crucial type.


2. Identifying Points of Discontinuity: For a given function, determine all points where it is discontinuous and classify the type of discontinuity.


3. Continuity of Composite Functions: Analyzing the continuity of f(g(x)) or g(f(x)), especially when one or both of f and g involve special functions or piecewise definitions.


4. Graphical Interpretation: Understanding continuity from the graph of a function.

JEE vs. CBSE: While CBSE board exams focus on the fundamental definition and simpler piecewise functions, JEE delves into more complex functions, multiple unknown constants, nested functions, and often requires advanced limit evaluation techniques (like L'Hopital's rule or series expansion, indirectly).

Example Problem (JEE Type)

Find the values of 'a' and 'b' such that the function f(x) is continuous for all x ∈ R:

        x2/a,        if 0 ≤ x < 1

f(x) =  a,              if x = 1

        b - 2/x2,      if x > 1

Solution Approach:
The function is polynomial in its defined intervals (except at x=0 or x=1 in the last case). We need to check continuity at the 'junction points' x=0 and x=1.

1. At x = 0:

LHL is not applicable as the domain starts from 0.

f(0) = 02/a = 0.

limx→0+ f(x) = limx→0+ (x2/a) = 0.

For continuity at x=0, f(0) = limx→0+ f(x), which is 0 = 0. This holds true for any non-zero 'a'. So, no specific condition on 'a' from x=0 directly.

2. At x = 1:

f(1) = a (given).

limx→1- f(x) = limx→1- (x2/a) = 12/a = 1/a.

limx→1+ f(x) = limx→1+ (b - 2/x2) = b - 2/12 = b - 2.

For continuity at x=1, LHL = RHL = f(1):

1/a = a => a2 = 1 => a = ±1.

a = b - 2

Case 1: a = 1

1 = b - 2 => b = 3.

So, (a, b) = (1, 3) is a valid solution.

Case 2: a = -1

-1 = b - 2 => b = 1.

So, (a, b) = (-1, 1) is a valid solution.

Always practice evaluating limits at critical points and applying the definition rigorously. This ensures accuracy in parameter finding questions.